The regular refinement of a tetrahedron called the ``father'' is done by subdividing it into smaller subtetrahedra called ``sons''. The first step is to find the midnodes of all edges of the father and to connect those on a common face. Then the four sons at the corners, which are obviously congruent with their father and have equal volume, can be ``cut off''. The remainder, an octahedron, has to be divided again, but this can be done in several ways. It can be cut along three different planes, which have the shape of parallelograms. Cutting along two of them generates four sons which are no longer congruent with the father. This strategy is equivalent to the insertion of one edge, namely the cutting edge of the above planes. However, the wrong choice can lead to degenerated elements whereas the choice of the shortest of the three possible edges will minimize the maximum measure of degeneracy of the sons ([Bey95], p. 362). If the initial elements have non-obtuse faces, this strategy produces at most three congruence classes.